Method for Direct Measurement of Structural Rolling ...glass.ruc.dk/pdf/articles/2020_TRR.pdf · When driving at constant speed, the fuel consumption goes into overcoming driving

Research Article

Transportation Research Record1–10� National Academy of Sciences:Transportation Research Board 2020Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/0361198120915699journals.sagepub.com/home/trr

Method for Direct Measurementof Structural Rolling Resistancefor Heavy Vehicles

Natasja R. Nielsen1, Karim Chatti2, Christoffer P. Nielsen3, Imen Zaabar4,Poul G. Hjorth5, and Tina Hecksher1

AbstractIn this paper, a new in situ method for determining the structural rolling resistance (SRR), defined as the dissipated energycaused by deformation of the pavement when subjected to a moving load, is presented. The method is based on the relationbetween SRR and the slope of the deflection basin under a moving load. Using the Traffic Speed Deflectometer, the deflectionslope is measured at several positions behind and in front of the right rear-end tire pair of a full-size truck trailer while drivingunder realistic conditions. The deflection slope directly under the tire is estimated from a linear interpolation between thetwo nearest sensors. A set of data from a test road segment located in Denmark is analyzed and the SRR coefficients arefound to be in the range 0.005% to 0.05%. The deflection slope measurements have a high reproducibility (repeated measure-ments agree within standard deviations of 4% to 10%) with high spatial resolution, and the method for calculating SRR fromthese measurements has the clear advantage that it requires no knowledge or model of the pavement structure or viscoelas-tic properties. Numerical simulations of pavement response show that the proposed interpolation method tends to underes-timate the actual SRR, and better estimates can be obtained by other interpolation schemes.

When driving at constant speed, the fuel consumptiongoes into overcoming driving resistance. Many differentfactors contribute to the driving resistance in a vehicle;among the most prominent are uphill driving, air drag,internal friction, and rolling resistance (1). It is estimatedthat for heavy trucks, 15% to 30% of the fossil fuel inputis used to overcome the rolling resistance (2). Rollingresistance losses arise from two main sources: 1) viscoe-lastic effects in the tires and 2) effects of the pavement,including unevenness, texture, and viscoelastic deforma-tion of the pavement (3–5). The focus in this paper is onthe latter.

An elastic or viscoelastic pavement subject to a mov-ing vehicle will deform underneath the tires. If the pave-ment is viscoelastic, this deformation will result in energydissipating into the pavement structure. The lost energyhas to be compensated through additional work from thevehicle engine, to maintain a constant driving speed (6).The amount of additional energy needed depends on thestructure of the pavement and this will be referred to asstructural rolling resistance (SRR) throughout the paper.

The deflection basin under a moving tire (z(x)) isasymmetric because of the viscoelastic properties of thepavement causing a time delay in the deflection of a

viscoelastic pavement. This time delay makes the maxi-mum deflection appear behind the center of the tire, asseen on Figure 1a. This means that the tire always willbe on an uphill slope (∂z(x= 0)

∂x.0), (see Figure 1b) and

thus has to do work in order to maintain a constantdriving speed (7). Using this uphill slope notion, theSRR can be calculated directly from the asymmetricdeflection basin (1, 8, 9). Deflection of a structure subjectto a moving load has been reported in the literature sincethe 1960s; for example, in (7), the viscoelastic response ofa Kelvin beam is analyzed, and the viscoelastic effectsreported to manifest themselves through an asymmetricdeflection basin.

1Department of Science and Environment, Roskilde University, Roskilde,

Denmark2Department of Civil and Environmental Engineering, Michigan State

University, East Lansing, MI3Greenwood Engineering A/S, Brøndby, Denmark4Department of Computer Science and Engineering, Michigan State

University, East Lansing, MI5DTU Compute, Technical University of Denmark, Lyngby, Denmark

Corresponding Author:

Natasja R. Nielsen: [email protected]

Although SRR has been studied for decades, it hasproven difficult to devise accurate and robust ways ofmeasuring it (10). As a consequence, little is knownabout the absolute magnitude of SRR or its relative con-tribution to the overall rolling resistance. Indirect mea-surements of the influence of the dissipative effects inbituminous layers have been estimated by comparingfuel consumption measurements on flexible and rigidpavements. These studies rely on the assumption thatrigid pavements have little or no viscous losses and thusthe difference in fuel consumption between these types ofpavements can be ascribed to the viscous behavior of theasphalt (8, 11, 12). However, it can be difficult to isolatethe effects that relate to the pavement structure fromother effects caused by, for example, texture or uneven-ness (9). In addition, unlike texture and unevenness, theeffect from pavement structure is found to be highlydependent on external parameters such as temperature,pavement conditions, and so forth (13). It is thereforedifficult to say anything conclusive on SRR influence onfuel consumption based on these types of experiments.

Direct estimates of SRR typically come from simula-tions of pavement deflections with pavement parametersobtained either from backcalculations using fallingweight deflectometer tests or other rheological measure-ments of the bituminous layer. An often used method isto simulate the pavement response in a finite pavementsection, as a moving load is passing with constant speed(14). From the response, one can obtain the displacementfield of the pavement surface and calculate the dissipatedenergy in the pavement (3, 10, 12, 15, 16). On the basisof such calculations, it is believed that the SRR loss issmaller than the energy loss caused by pavement textureand unevenness (15), but whether it is negligible or

significant enough that it should be included in pavementplanning is not clear.

Development of methods for reliable measurement ofthe pavement’s influence on the vehicle fuel consumptionis thus highly desirable when making lifecycle assessmentstudies of pavements and should be included in the devel-opment of sustainable pavement designs (3, 6).

This paper presents a novel method for determiningthe SRR under realistic driving conditions using theTraffic Speed Deflectometer technology developed byGreenwood Engineering. The technique measures theslope of the deflection basin between the right pair ofrear-end tires of a full-size truck trailer, as it moves atrealistic driving speeds. Thus, the uphill slope seen by thetire, which is caused by the deformation of the pavement,is directly measured and, from this, the associated SRRloss can be calculated. The estimated SRR is thusobtained under conditions directly comparable to whatnormal traffic experiences. The method gives spatiallyresolved (10-m resolution), reproducible, and robust esti-mates of SRR, even in road segments where the valuefluctuates considerably, making it a reliable and model-free method to measure SRR.

Aim

The aim of the paper is to present a new concept for mea-suring SRR using Traffic Speed Deflectometer (TSD)technology. The TSD measures the slope of the deflec-tion basin under the tires of a truck trailer during driv-ing. The concept and its robustness are demonstrated bypilot measurements of a test road segment of 9 km, andthe underlying assumptions are discussed in the light ofnumerical pavement simulations.

(a) (b)

Figure 1. (a) Simulated deflection basin underneath a moving load for an elastic (solid line) and viscoelastic (dotted line) pavement, and(b) associated deflection slope for the elastic and viscoelastic pavement. The basin is obtained using a numerical simulation explained at alater point in this paper.

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The TSD Concept

The TSD is conventionally used for continuous bearingcapacity measurements by evaluating the slope of thepavement deflection basin. It has the advantage that itmakes continuous measurements of the deflection slopeand that the TSD trailer is a normal truck trailer andthus can measure under normal driving speed and loadas well as measuring directly in the wheel path. In thisstudy a full axle load of 10 tonnes was used.

The TSD device measures the deflection velocity ofthe pavement as it is subjected to a moving load. This isdone by use of Doppler lasers that measure the verticalvelocity of the pavement (see Figure 2b). The TSD truckis equipped with nine Doppler lasers (sensors): three sen-sors located behind and six in front of the rear-end axle,as shown in Figure 2a. Their exact positions relative tothe center of the axle (in meters) are

Sensor position= ½�0:366, � 0:269, � 0:167,

0:163, 0:260, 0:362, 0:662, 0:964, 1:559� :ð1Þ

The measured pavement velocity is adjusted such thateffects caused by vertical movements of the truck are sub-tracted. This is done by using a reference laser mounted3.1m from the rear-end axle, where the deflection of thepavement is assumed zero (red sensor on Figure 2a). Thetechnique is explained in more detail in (17–20).

Figure 2b shows how the vertical pavement velocity(vd) is measured in a given point using a Doppler laser.The deflection slope at that point ( ∂z

∂x) corresponds to the

slope of the tangent going through the point (gray dottedline) and can be found by dividing vd by the horizontaldriving speed (v),

∂z

∂x=

∂z∂t∂x∂t

=vd

vð2Þ

The driving speed, v, is measured using an odometerlocated behind the right rear-end tire pair.

Deflection Slope Data

For this study, three repeated measurements were madewith the TSD, on a 9.7-km road section nearCopenhagen, Denmark. The measurements were con-ducted in the spring of 2018 with almost constant airtemperature (;148C) and road temperature (;188C)throughout all three measurements. The driving speedwas between 50 km/h and 60 km/h; the exact drivingspeed was recorded continuously during all measure-ments. The measured deflection slopes for each sensorwere collected at a sampling frequency of 1,000 samplesper second and subsequently averaged over 10m. A plotof the mean value for the three subsequent measurementsof each sensor as a function of the driven distance is seenon Figure 3. The measured deflection slope for each sen-sor varies significantly throughout the measured dis-tance. This variation is however highly reproducible,with average standard deviations between 12 mm=m and26 mm=m (corresponding to 4-10%) between the threemeasurement runs.

The inset in Figure 3 shows the measured deflectionslope as a function of the sensor position measured at4.5 km (marked in gray in the main image). The center ofthe axle in this plot is at x= 0, indicated with a blackdotted line. As mentioned in the introduction, the deflec-tion slope curve is characterized by the minimum deflec-tion slope occurring behind and the maximum deflectionslope in front of the tire. The asymmetry in minimumand maximum peak magnitudes is believed to be causedby damping in the pavement. Thus, the location andmagnitude of the maximum and minimum carries infor-mation about the viscoelastic properties of the pavement.

For analysis of the data, it is necessary to estimate thedeflection slope at the axle location, that is, around

(a)

(b)

Figure 2. (a) Top view sketch of the Traffic Speed Deflectometer.Nine Doppler lasers are located in between the right rear-end tirepair, as indicated with blue dots. Note that the drawing is not toscale, and the tires in the tire-pair are only separated by 64 mm. (b)Vertical pavement velocity (vd) at a given point measured using aDoppler laser. See text for further details.

Nielsen et al 3

x= 0, where a measurement cannot be taken becauseof the presence of the axle. Instead, the slope must beinferred from the measured locations in front of andbehind the center position. This task is easier when thefeatures of the deflection slope are fully captured bythe sensors, which is not the case for all traces.Accordingly, the measurements were partitioned intothree groups based on the behavior of the signal inSensors 4, 5, and 6 (Table 1), which gives an indicationof where the maximum is located: Group 1 was usedfor measurements for which the maximum was notcaptured by the sensors and therefore had to belocated closer to the center of the axle than Sensor 4;Group 2 was used for measurements for which themaximum was partly captured by the sensors; andGroup 3 was used for measurements for which themaximum was fully captured by the sensors (see Table1). Examples of measurements from each group areshown in Figure 4a. Here the symbols are the averagevalues of the three repeated measurements and theerrorbars represent the standard deviations, showing ahigh degree of reproducibility. Within Groups 1 and 2,a big variation was found in the magnitude of the

maximum and the minimum, whereas for measure-ments belonging to Group 3 this variation was notobserved.

Calculating the SRR

This section shows how the SRR loss can be calculateddirectly from the measured deflection slope data. In thefollowing it is assumed that the applied load is a pointload at the center of the tire, corresponding to x= 0 andwith the magnitude FL. The dissipated power caused bySRR, PSRR, can be found from the applied load and thepavement velocity at this point,

PSRR =FLvd(x= 0)=FLv∂z

∂x(x= 0) ð3Þ

where the last equality sign comes from Equation 2.In the case of a perfectly elastic pavement, the maxi-

mum deflection will occur directly under the load, mak-ing the deflection slope at this point zero and thusPSRR= 0. For a viscoelastic pavement, however, themaximum deflection occurs behind the load and there isan uphill slope underneath the load, thus PSRR.0, as

Figure 3. Measured deflection slope for each sensor as a function of the distance, with inset showing a plot of the deflection slopesmeasured at 4.5 km as a function of sensor location.

Table 1. Partitioning of the Traffic Speed Deflectometer Measurements into Groups

Group Behavior of signal in Sensors 4, 5, and 6 Location of maximum

Group 1 Monotonic decrease Closer to center of axle than Sensor 4Group 2 Increasing or equal from Sensors 4 to 5 and then decreasing in Sensor 6 Partly captured by Sensors 4 and 5Group 3 Monotonic increase Fully captured by the sensors

Note: The division is made based on the behavior of the measured deflection slope in Sensors 4, 5, and 6. In total this gives three groups, illustrated in

Figure 4a.

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already illustrated in Figure 1. Note that, the deflectionmaximum occurs behind the center of the load, whethera point load or a finite contact area is considered. Thus,the tire also experiences an uphill slope if considered afinite contact area, and thereby has PSRR.0 wheneverthere is damping in the pavement.

To estimate the deflection slope directly under the tire,a linear interpolation is used between the measureddeflection slope in the two sensors located closest to thecenter (Sensors 3 and 4), located at x= � 0:167 m andx= 0:163 m respectively, as shown in Figure 4b.Therefore the dissipated energy can be written as

PSRR=FLvb ð4Þ

where b is the intersection of the linear interpolation∂z∂x(x)= ax+ b with the z -axis, ∂z

∂x(x= 0). From the dissi-

pated power, the rolling resistance force can be definedas FSRR= Psrr

v=FLb. Using the standard definition of

rolling resistance coefficient as the ratio between rollingresistance force and load, this leads to the following sim-ple relation between deflection slope at x= 0 and theSRR coefficient:

CSRR=FSRR

FL

= b ð5Þ

Using these relations on the data trace presented inFigure 4b, we find an SRR power of 49 W6 6 W, anSRR force of 6:8 N6 0:8 N, and CSRR= 1:4 � 10�4 6

1:6 � 10�5 or 0:014%6 0:0016%.The CSRR values for all measurement sets were found

following this procedure, and the results are presented inFigure 5. Here, the different groups are marked with

different colors, the symbols represent the mean valuesof the three repeated measurements, and the error barsare found as the standard deviation of the three measure-ments. We see that the CSRR value varies considerablyover the traveled distance, from 0.005% to 0.05%, withmost data points in the region from 0.01% to 0.02%.The method shows a good reproducibility with low stan-dard deviations, even in regions where the CSRR changesrapidly with distance. This demonstrates that the methodis robust and can measure the CSRR values of the roadprecisely, with high spatial resolution even under chang-ing pavement conditions.

The different data groups are indicated with red,green, and blue on Figure 5. Average values of PSRR,FSRR and CSRR for each group are shown in Table 2.The groups were divided based on the location of themaximum, captured by Sensors 4, 5, and 6, and it is pos-sible to see a clear difference in the SRR values withinthe different groups. Furthermore, the variations inCSRR, with distance seen in Figure 5 follow the trendsseen in the measured deflection slopes in Figure 3. Thisis because a large deflection slope signal in the sensorsclosest to the axle (Sensors 3 and 4) generally results in ahigh intersection value with the y� axis, and thus a highcalculated CSRR (Equation 5).

The magnitude and the location of the peaks in thedeflection slope curves are determined by the shape ofthe deflection basin, which mainly is controlled by therelative stiffness of the top asphalt layer compared withthe lower layers. For situations with a relatively stifftop layer, the deflection basin will be broad and have asmall amplitude, resulting in curves like those of Group3 and a small SRR. A relatively soft top layer, on the

(a) (b)

Figure 4. (a) Representative examples of deflection slope plotted as a function of sensor location for the different measurement groups(see Table 1), and (b) linear interpolation between the measured values in the two sensors closest to the axle for data measured at 4.5 kmbelonging to Group 2.

Nielsen et al 5

other hand, will give a deep and narrow basin, givingdeflection curves like those of Group 1 and a higherSRR. This is consistent with what is visible in themeasurements.

Impact of a Finite Contact Surface

For the calculations of the dissipated power and CSRR

above, it is assumed that the interaction between tire androad can be described as a point load. This is a simplifica-tion of the real interaction between the tire and the pave-ment where the contact surface has a finite area. Toinvestigate whether this approximation has a significantinfluence on the calculated SRR loss, an expression isadopted for the power dissipation derived by (9). Theexpression is based on a moving reference frame withconstant velocity, which is consistent with the TSD setup.Furthermore, it is assumed that the tire is elastic andtherefore does not dissipate energy and that the tire pro-vides a uniform applied stress to the surface,

Pcontact areaSRR = pv

ðS

∂z(X , y, z)

∂XdS ð6Þ

wherep is the tire pressure,v is the driving speed,Z is the vertical component of the displacement field

of the pavement surface, and∂z(X , y, z)

∂Xis the deflection slope.

The integral is taken over the contact surface, S, which isthe area where the tire is in contact with the pavement.Plugging in a linearly varying deflection slope and assum-ing a circular contact area obtains

Pcontact areaSRR = pv

ðS

∂z(X , y, z)

∂XdS

= pv

ðr

�r

ð ffiffiffiffiffiffiffiffiffiffir2�X 2p

�ffiffiffiffiffiffiffiffiffiffir2�X 2p (aX + b) dy dX

= pvbpr2 =Fvb=Ppoint loadsrr

ð7Þ

Figure 5. Calculated CSRR values plotted versus distance, with insets showing a steady and a varying section with standard deviationsillustrated by error bars. The different colors represent the three different Groups (see Table 1).Note: CSRR = structural rolling resistance coefficient.

Table 2. Average CSRR, FSRR, and PSRR for the Three Groups of Traffic Speed Deflectometer Data

Group CSRR [–] PSRR (W) FSRR (N) # in group

Group 1 1:7 � 10�4 6 6 � 10�5 124:2 6 57 8:6 6 3:0 506Group 2 1:2 � 10�4 6 4 � 10�5 84:9 6 30 5:9 6 1:8 272Group 3 0:9 � 10�4 6 3 � 10�5 61:7 6 21 4:2 6 1:3 159

Note: It can be seen that SRR for data in Group 1 is largest, followed by Group 2, and then Group 3. The number of measurements within the dataset

belonging to each group is listed in the last column. CSRR = SRR coefficient; FSRR = rolling resistance force; PSRR = dissipated power due to SRR; SRR =

structural rolling resistance.

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Thus, for a linearly varying deflection slope the powerdissipated over a finite contact area is equal to the powerdissipated at a point load.

Model Calculation of Pavement Response

So far, it has been assumed that the deflection slopeunderneath the tire is linear and can be found by interpo-lation between the two sensors nearest to the axle. Thevalidity of this assumption will now be investigated byuse of simulated deflection slopes. The purpose of this issolely to generate curves with similar behaviors to thoseobserved in the measurements, and to investigate howwell the assumption of a linear deflection slope performsfor the simulated curves. In particular, this is not anattempt to model the exact pavement response measured,but rather a theoretical exploration of the interpolationapproach.

For simulating the pavement response, the investiga-tion uses the time-domain based viscoelastic solverViscoWave II-M, developed at Michigan StateUniversity (21, 22). ViscoWave II-M employs the so-called spectral element method to solve the wave propa-gation problem in the pavement structure and calculatethe pavement response to an arbitrary loading. Themodel can simulate the time-dependent responses andallows each pavement layer to be either elastic or viscoe-lastic (23).

The program was modified slightly for this study suchthat the simulated conditions are similar to the TSDsetup and therefore can be used for comparison. Theoriginal solver calculates the pavement deflection underthe tire in a steady reference frame. The modified versioncalculates the response between the two tires in the tirereference frame, that is, a moving reference frame. Fromthe simulated deflection curve the corresponding slope iscalculated and filtered to remove numerical noise.

The pavement structure used for the simulation con-sists of three layers, representing an asphalt layer, a baselayer, and a subgrade layer. Four different pavementmodels with identical construction are simulated, onlychanging viscoelastic parameters for the asphalt (top)layer. The parameters for the structure (height, elasticmoduli, Poisson’s ratio and density) are chosen to betypical values for these kinds of pavement layers andthey are listed in Table 3. The viscoelastic properties ofthe asphalt layer are described by the relaxation modulusE(t), given by

log (E(t))= c1 +c2

1+ e(�c3�c4 log (tR))ð8Þ

log (tR)= log (t)� log (aT ) ð9Þ

where

c1, . . . , c4 are the sigmoid coefficients,tR is the reduced time, andaT is the shift factor (16).The parameters for the relaxation moduli are taken

from backcalculated falling weight deflectometer tests onroad segments located in California, to have realistic E(t)curves (16). The characteristics of these moduli rangefrom very stiff with high damping to very soft with littledamping (see Table 4). These sets of parameters generateddeflection slope curves with a similar variation to thatseen in the data groups as shown in Figure 6a. In thesimulated deflection curves, the stiff pavement with largedamping (PAV4) shows a small deflection and deflectionslope peaks far apart, whereas the soft pavement with lit-tle damping (PAV1) has the opposite behavior. Probably,other choices of pavement parameters could result in simi-lar deflection basins. However, for the present purposethe detailed input parameters of the model are not soimportant, as long as they are reasonably realistic.

In Figure 6b a zoom of the contact region for each ofthe simulated deflection slope curves is shown. The con-tact area between the tire and pavement is assumed circu-lar with radius (r) and the interval [-r; r] is marked withgray color. The idea is to determine how much the actualSRR in the simulated deflection slope curve deviates from

Table 3. Mechanical Characteristics for the Simulated Pavement

Asphalt Base Subgrade

E(t) E2 = 124.3 MPa E3 = 65.4 MPan = 0.35 n = 0.35 n = 0.45

r = 2,322.7kg

m3r = 2,082.4

kg

m3r = 1,762

kg

m3

h= 0.15 m h = 0.3 m h = N

Note: All pavement structures are made of three layers, each characterized

by their Poisson’s Ratio (n), mass density (r), average thickness (h) and the

relaxation modulus (E). The relaxation modulus for the asphalt layer is

given by Equation 8.

Table 4. Properties of the Four Different E(t) Used for the Study

Properties

Pavements

PAV1 PAV2 PAV3 PAV4

Sigmoid coefficientsc1 1.4 1.054 0.978 1.67c2 2.04 2.986 3.8 3.39c3 0.944 0.335 0.521 0.981c4 –0.417 –0.436 –0.519 –0.767Shift factor log (aT) 0.37 0.32 0.49 0.34E(t) characteristicsE0 [MPa] 2,753 10,956 59,970 114,820E0 � E‘ [MPa] 2,728 10,945 59,960 114,770Stiffness ����������������������������!Amount of damping ����������������������������!

Nielsen et al 7

the SRR obtained by assuming a linear interpolationbetween coordinates of the two sensors closest to the axlein the measurement. The linear interpolation is markedon Figure 6b by a black line and the SRR is found as theintersection of this linear interpolation with the z -axis.

Calculating the SRR for the simulations involves inte-grating the deflection slope over the contact area asdescribed above in Equation 6, again assuming a circularcontact area with origin in x= 0 and radius r. A value ofr= 14:5 cm, found from the tire pressure and axle loadof the TSD, is used.

In addition to the linear interpolation, a cubic splineinterpolation was also created. In this, a 3rd order poly-nomial is used to find the values in between the two inter-polation points instead of a linear function, thus giving asmoother interpolation curve. As this method has moreunknown parameters to fit than the linear, nine simula-tion points are used, corresponding with the coordinatesof the TSD sensors, to make the interpolation. The splineinterpolation is marked on Figure 6b with a dotted line.The spline interpolation is included in an attempt toapproximate the actual deflection slope in the contactarea better.

The relative difference between the interpolations andthe simulation curves is found by the relative differencein the dissipated energy over the contact area,

DPint:

P=

ÐS∂zsim

∂xdS �

ÐS∂zint:

∂xdSÐ

S∂zsim

∂xdS

ð10Þ

The calculated PSRR values for the different deflectionslope curves and the two interpolated curves are listed inTable 5 along with their relative differences.

The analysis shows that the difference between thesimulated deflection slope and the linear interpolation issmall for PAV4, DPlinear

P= 9%, where the deflection maxi-

mum and minimum are far apart. With decreasing stiff-ness, and thus smaller distance between maximum andminimum, the error increases, with the largest deviationfound in PAV1, where DPlinear

P= 49%.

The spline interpolation shows the same trend, but itgives a better estimate of SRR. Thus, for the PAV1 thedifference is only DPspline

P= 23%, whereas for PAV4 it

gives practically the same value as the model curve.It can be concluded that the linear assumption is valid

when the deflection slope peaks are far apart, whereas it

(a) (b)

Figure 6. (a) Simulated deflection and deflection slope curves for four pavements with different E(t) of the asphalt layer, and (b) close upof the simulated deflection slope curves with a linear and a cubic spline interpolation. The contact area interval is marked with gray color.

Table 5. Calculated Change in PSRR of the Simulated DeflectionSlope and the Linear and Cubic Spline Interpolations for DifferentPavements, Also Showing Values for the Calculated PSRR of Boththe Simulation and the Interpolations for Each Pavement

Pavement PAV1 PAV2 PAV3 PAV4

Psimsrr 335 W 356 W 170 W 59 W

Plinearsrr 172 W 220 W 134 W 54 W

Psplinesrr 257 W 297 W 159 W 58 W

DPlinear

P49% 38% 17% 9%

DPspline

P 23% 17% 6% 2%

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underestimates SRR when the peaks are too close to theorigin to be resolved. The spline interpolation in all casesgives a slightly better estimate of SRR, especially forpavements for which the peaks are close together.

Lastly, the numerical calculations were employed to esti-mate the difference in the deflection slope obtained under-neath the tires and at the location of the TSD sensors. Inthe TSD setup, the sensors are located between the tire pair(see Figure 2a) and therefore the deflection slopes reportedin this paper are measured in between the tire pair. Thisdeviates from the analysis assumptions about the contactarea in Equation 6, where it is assumed to be circular withorigin in x = 0. By simulating the pavement deflection forpavement PAV1 directly underneath the tires and inbetween the tire pair, respectively, it was found that the dif-ference in PSRR is 3.6%. Consequently, this does not havea significant impact on the final SRR results.

Summary and Outlook

This paper has presented a model-free way to estimateSRR from pavement deflection slope measurementsobtained with the TSD. In the simplest approach, it wasassumed that the contact between tire and road is point-like (i.e., a ‘‘moving point load’’). In that case, the SRRcoefficient, CSRR, is simply given as the value of thedeflection slope curve at the position of the point load.Because it is not possible to measure exactly at that posi-tion because of the presence of the axle, the deflectionslope was estimated from a linear interpolation of nearbymeasurement points behind and in front of that location.The point load assumption is shown to be equivalent tocalculations based on a finite contact area, if the deflec-tion slope varies linearly within the contact region.

A set of data from a test road was investigated andthe values of CSRR found by this method span from0:005 % to 0:05 %, which are modest values comparedwith typical tire rolling resistance coefficients that are inthe range 0.5% to 1%. The values are slightly lower thanthose found in empirical and numerical studies on thesubject (9–11, 15). The data were divided into threegroups based on how much of the deflection slope maxi-mum was resolved by the TSD sensors. This was basedon the hypothesis that this criterion is critical for the lin-ear interpolation to be a good estimate of the deflectionslope under the tire. It was found that for measurementsin Group 1 with maximum located closest to the load,the SRR was highest, and for Group 3 with maximumlocated the furthest away, the SRR was lowest. Throughsimulated deflection slope curves obtained using the pro-gram ViscoWave II-M the linear interpolation was foundto underestimate the actual SRR by up to ; 50% in theworst case. Using a cubic spline interpolation betweennine positions corresponding to the TSD sensor positionsimproved the SRR estimate considerably, confirming

that the resolution of the maximum is critical for the lin-ear interpolation approach to give accurate results.Further development of the interpolation method willimprove the method and improve the accuracy of theestimated SRR values. By use of numerical studies theauthors aim to develop a simple functional expressionthat will allow the deflection slope values underneath theaxle to be estimated with greater accuracy.

The strength of the method is that it requires noknowledge about the pavement structure or pavementproperties. Furthermore, the use of the TSD vehiclemakes data collection relatively fast and easy and thedeflection slope measurements are very precise. Thisleads to reproducible values of CSRR determined withlow standard deviation, even in areas of the road wherethe values vary considerably.

The measurements included in this study were madeon a test road with the purpose of illustrating the newmethod and this was chosen for purely practical reasons.They were carried out in relatively cold conditions (pave-ment temp. ;188C) and a future study with higher pave-ment and air temperature is expected to provide higherSRR values. In the study, it was found that the magni-tude and location of the maximum deflection slope iscorrelated with the SRR. It is expected that these quanti-ties are mainly dependent on the relative stiffness of thetop layer compared with the underlying layers and thatthe location of the maximum deflection depends on theamount of damping in the pavement (damping in toplayer, foundation, or a combination). The relationshipbetween these pavement characteristics and the behaviorof the deflection slope curve should be explored furtherby use of simple physical models.

Through this new, easy method for measuring SRR, itwill be feasible to conduct a series of tests on roads withdifferent pavement structures and thus investigate therelationship between pavement structure and SRR.Furthermore, the impact of road temperature or drivingspeed could also be investigated. Such large-scale sys-tematic surveys could provide much needed clarity in thestudy of SRR, and establish under which circumstancesSRR is important for overall fuel consumption as well ashow it is affected by various parameters.

Author Contributions

The authors confirm contribution to the paper as follows: studyconception and design: Natasja R. Nielsen, Christoffer P.Nielsen, Tina Hecksher, Poul G. Hjorth; data collection:Christoffer P. Nielsen; analysis and interpretation of results:Natasja R. Nielsen, Christoffer P. Nielsen, Imen Zaabar,Karim Chatti; draft manuscript preparation: Natasja R.Nielsen, Christoffer P. Nielsen, Tina Hecksher, Poul G. Hjorth,Karim Chatti, Imen Zaabar. All authors reviewed the resultsand approved the final version of the manuscript.

Nielsen et al 9

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The authors received no financial support for the research,authorship, and/or publication of this article.

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